# Properties

 Label 507.2.e.e Level $507$ Weight $2$ Character orbit 507.e Analytic conductor $4.048$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7} - \beta_{3} q^{8} + (\beta_1 - 1) q^{9}+O(q^{10})$$ q - b2 * q^2 + b1 * q^3 + (b1 - 1) * q^4 + (b3 - b2) * q^6 + (-2*b3 + 2*b2) * q^7 - b3 * q^8 + (b1 - 1) * q^9 $$q - \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{6} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{7} - \beta_{3} q^{8} + (\beta_1 - 1) q^{9} - 2 \beta_{2} q^{11} - q^{12} + 6 q^{14} + 5 \beta_1 q^{16} + (6 \beta_1 - 6) q^{17} + \beta_{3} q^{18} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{19} - 2 \beta_{3} q^{21} + (6 \beta_1 - 6) q^{22} - \beta_{2} q^{24} - 5 q^{25} - q^{27} - 2 \beta_{2} q^{28} - 6 \beta_1 q^{29} - 2 \beta_{3} q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} + (2 \beta_{3} - 2 \beta_{2}) q^{33} + 6 \beta_{3} q^{34} - \beta_1 q^{36} + 4 \beta_{2} q^{37} + 6 q^{38} + 4 \beta_{2} q^{41} + 6 \beta_1 q^{42} + (4 \beta_1 - 4) q^{43} + 2 \beta_{3} q^{44} - 2 \beta_{3} q^{47} + (5 \beta_1 - 5) q^{48} - 5 \beta_1 q^{49} + 5 \beta_{2} q^{50} - 6 q^{51} + 6 q^{53} + \beta_{2} q^{54} + ( - 6 \beta_1 + 6) q^{56} - 2 \beta_{3} q^{57} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{58} + (6 \beta_{3} - 6 \beta_{2}) q^{59} + ( - 2 \beta_1 + 2) q^{61} + 6 \beta_1 q^{62} - 2 \beta_{2} q^{63} + q^{64} - 6 q^{66} + 6 \beta_{2} q^{67} - 6 \beta_1 q^{68} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{71} + (\beta_{3} - \beta_{2}) q^{72} + ( - 12 \beta_1 + 12) q^{74} - 5 \beta_1 q^{75} - 2 \beta_{2} q^{76} + 12 q^{77} - 8 q^{79} - \beta_1 q^{81} + ( - 12 \beta_1 + 12) q^{82} - 2 \beta_{3} q^{83} + (2 \beta_{3} - 2 \beta_{2}) q^{84} + 4 \beta_{3} q^{86} + ( - 6 \beta_1 + 6) q^{87} + 6 \beta_1 q^{88} - 4 \beta_{2} q^{89} - 2 \beta_{2} q^{93} + 6 \beta_1 q^{94} + 3 \beta_{3} q^{96} + (8 \beta_{3} - 8 \beta_{2}) q^{97} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{98} + 2 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^2 + b1 * q^3 + (b1 - 1) * q^4 + (b3 - b2) * q^6 + (-2*b3 + 2*b2) * q^7 - b3 * q^8 + (b1 - 1) * q^9 - 2*b2 * q^11 - q^12 + 6 * q^14 + 5*b1 * q^16 + (6*b1 - 6) * q^17 + b3 * q^18 + (-2*b3 + 2*b2) * q^19 - 2*b3 * q^21 + (6*b1 - 6) * q^22 - b2 * q^24 - 5 * q^25 - q^27 - 2*b2 * q^28 - 6*b1 * q^29 - 2*b3 * q^31 + (3*b3 - 3*b2) * q^32 + (2*b3 - 2*b2) * q^33 + 6*b3 * q^34 - b1 * q^36 + 4*b2 * q^37 + 6 * q^38 + 4*b2 * q^41 + 6*b1 * q^42 + (4*b1 - 4) * q^43 + 2*b3 * q^44 - 2*b3 * q^47 + (5*b1 - 5) * q^48 - 5*b1 * q^49 + 5*b2 * q^50 - 6 * q^51 + 6 * q^53 + b2 * q^54 + (-6*b1 + 6) * q^56 - 2*b3 * q^57 + (-6*b3 + 6*b2) * q^58 + (6*b3 - 6*b2) * q^59 + (-2*b1 + 2) * q^61 + 6*b1 * q^62 - 2*b2 * q^63 + q^64 - 6 * q^66 + 6*b2 * q^67 - 6*b1 * q^68 + (-2*b3 + 2*b2) * q^71 + (b3 - b2) * q^72 + (-12*b1 + 12) * q^74 - 5*b1 * q^75 - 2*b2 * q^76 + 12 * q^77 - 8 * q^79 - b1 * q^81 + (-12*b1 + 12) * q^82 - 2*b3 * q^83 + (2*b3 - 2*b2) * q^84 + 4*b3 * q^86 + (-6*b1 + 6) * q^87 + 6*b1 * q^88 - 4*b2 * q^89 - 2*b2 * q^93 + 6*b1 * q^94 + 3*b3 * q^96 + (8*b3 - 8*b2) * q^97 + (-5*b3 + 5*b2) * q^98 + 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^9 $$4 q + 2 q^{3} - 2 q^{4} - 2 q^{9} - 4 q^{12} + 24 q^{14} + 10 q^{16} - 12 q^{17} - 12 q^{22} - 20 q^{25} - 4 q^{27} - 12 q^{29} - 2 q^{36} + 24 q^{38} + 12 q^{42} - 8 q^{43} - 10 q^{48} - 10 q^{49} - 24 q^{51} + 24 q^{53} + 12 q^{56} + 4 q^{61} + 12 q^{62} + 4 q^{64} - 24 q^{66} - 12 q^{68} + 24 q^{74} - 10 q^{75} + 48 q^{77} - 32 q^{79} - 2 q^{81} + 24 q^{82} + 12 q^{87} + 12 q^{88} + 12 q^{94}+O(q^{100})$$ 4 * q + 2 * q^3 - 2 * q^4 - 2 * q^9 - 4 * q^12 + 24 * q^14 + 10 * q^16 - 12 * q^17 - 12 * q^22 - 20 * q^25 - 4 * q^27 - 12 * q^29 - 2 * q^36 + 24 * q^38 + 12 * q^42 - 8 * q^43 - 10 * q^48 - 10 * q^49 - 24 * q^51 + 24 * q^53 + 12 * q^56 + 4 * q^61 + 12 * q^62 + 4 * q^64 - 24 * q^66 - 12 * q^68 + 24 * q^74 - 10 * q^75 + 48 * q^77 - 32 * q^79 - 2 * q^81 + 24 * q^82 + 12 * q^87 + 12 * q^88 + 12 * q^94

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} ) / 3$$ (-b3 + 2*b2) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1
 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.866025 + 1.50000i 0.500000 0.866025i −0.500000 0.866025i 0 0.866025 + 1.50000i −1.73205 3.00000i −1.73205 −0.500000 0.866025i 0
22.2 0.866025 1.50000i 0.500000 0.866025i −0.500000 0.866025i 0 −0.866025 1.50000i 1.73205 + 3.00000i 1.73205 −0.500000 0.866025i 0
484.1 −0.866025 1.50000i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.866025 1.50000i −1.73205 + 3.00000i −1.73205 −0.500000 + 0.866025i 0
484.2 0.866025 + 1.50000i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 −0.866025 + 1.50000i 1.73205 3.00000i 1.73205 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.e.e 4
13.b even 2 1 inner 507.2.e.e 4
13.c even 3 1 507.2.a.f 2
13.c even 3 1 inner 507.2.e.e 4
13.d odd 4 1 507.2.j.a 2
13.d odd 4 1 507.2.j.c 2
13.e even 6 1 507.2.a.f 2
13.e even 6 1 inner 507.2.e.e 4
13.f odd 12 2 39.2.b.a 2
13.f odd 12 1 507.2.j.a 2
13.f odd 12 1 507.2.j.c 2
39.h odd 6 1 1521.2.a.l 2
39.i odd 6 1 1521.2.a.l 2
39.k even 12 2 117.2.b.a 2
52.i odd 6 1 8112.2.a.bv 2
52.j odd 6 1 8112.2.a.bv 2
52.l even 12 2 624.2.c.e 2
65.o even 12 2 975.2.h.f 4
65.s odd 12 2 975.2.b.d 2
65.t even 12 2 975.2.h.f 4
91.bc even 12 2 1911.2.c.d 2
104.u even 12 2 2496.2.c.d 2
104.x odd 12 2 2496.2.c.k 2
156.v odd 12 2 1872.2.c.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.b.a 2 13.f odd 12 2
117.2.b.a 2 39.k even 12 2
507.2.a.f 2 13.c even 3 1
507.2.a.f 2 13.e even 6 1
507.2.e.e 4 1.a even 1 1 trivial
507.2.e.e 4 13.b even 2 1 inner
507.2.e.e 4 13.c even 3 1 inner
507.2.e.e 4 13.e even 6 1 inner
507.2.j.a 2 13.d odd 4 1
507.2.j.a 2 13.f odd 12 1
507.2.j.c 2 13.d odd 4 1
507.2.j.c 2 13.f odd 12 1
624.2.c.e 2 52.l even 12 2
975.2.b.d 2 65.s odd 12 2
975.2.h.f 4 65.o even 12 2
975.2.h.f 4 65.t even 12 2
1521.2.a.l 2 39.h odd 6 1
1521.2.a.l 2 39.i odd 6 1
1872.2.c.e 2 156.v odd 12 2
1911.2.c.d 2 91.bc even 12 2
2496.2.c.d 2 104.u even 12 2
2496.2.c.k 2 104.x odd 12 2
8112.2.a.bv 2 52.i odd 6 1
8112.2.a.bv 2 52.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{4} + 3T_{2}^{2} + 9$$ T2^4 + 3*T2^2 + 9 $$T_{5}$$ T5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 12T^{2} + 144$$
$11$ $$T^{4} + 12T^{2} + 144$$
$13$ $$T^{4}$$
$17$ $$(T^{2} + 6 T + 36)^{2}$$
$19$ $$T^{4} + 12T^{2} + 144$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 6 T + 36)^{2}$$
$31$ $$(T^{2} - 12)^{2}$$
$37$ $$T^{4} + 48T^{2} + 2304$$
$41$ $$T^{4} + 48T^{2} + 2304$$
$43$ $$(T^{2} + 4 T + 16)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T - 6)^{4}$$
$59$ $$T^{4} + 108 T^{2} + 11664$$
$61$ $$(T^{2} - 2 T + 4)^{2}$$
$67$ $$T^{4} + 108 T^{2} + 11664$$
$71$ $$T^{4} + 12T^{2} + 144$$
$73$ $$T^{4}$$
$79$ $$(T + 8)^{4}$$
$83$ $$(T^{2} - 12)^{2}$$
$89$ $$T^{4} + 48T^{2} + 2304$$
$97$ $$T^{4} + 192 T^{2} + 36864$$